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Theorem nfcnv 5471
Description: Bound-variable hypothesis builder for converse. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
nfcnv.1 𝑥𝐴
Assertion
Ref Expression
nfcnv 𝑥𝐴

Proof of Theorem nfcnv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 5287 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2 nfcv 2907 . . . 4 𝑥𝑧
3 nfcnv.1 . . . 4 𝑥𝐴
4 nfcv 2907 . . . 4 𝑥𝑦
52, 3, 4nfbr 4858 . . 3 𝑥 𝑧𝐴𝑦
65nfopab 4879 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
71, 6nfcxfr 2905 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wnfc 2894   class class class wbr 4811  {copab 4873  ccnv 5278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-cnv 5287
This theorem is referenced by:  nfrn  5539  nfpred  5872  nffun  6093  nff1  6283  nfsup  8568  nfinf  8599  gsumcom2  18654  ptbasfi  21678  mbfposr  23724  itg1climres  23786  funcnvmpt  29938  nfwsuc  32228  aomclem8  38332  rfcnpre1  39854  rfcnpre2  39866  smfpimcc  41678
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